Extremal digraphs whose walks with the same initial and terminal vertices have distinct lengths

Let DD be a digraph of order nn in which any two walks with the same initial vertex and the same terminal vertex have distinct lengths. We prove that DD has at most (n+1)2/4(n+1)2/4 arcs if nn is odd and n(n+2)/4n(n+2)/4 arcs if nn is even. The digraphs attaining this maximum size are determined.